On an installation of Buchberger's algorithm
Journal of Symbolic Computation
An introduction to commutative and noncommutative Gro¨bner bases
Selected papers of the second international colloquium on Words, languages and combinatorics
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Groebner Bases in Non-Commutative Algebras
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
Computational Commutative Algebra 1
Computational Commutative Algebra 1
Letterplace ideals and non-commutative Gröbner bases
Journal of Symbolic Computation
Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra
Journal of Symbolic Computation
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Recently, the notion of "letterplace correspondence" between ideals in the free associative algebra KX and certain ideals in the so-called letterplace ring KXP has evolved. We continue this research direction, started by La Scala and Levandovskyy, and present novel ideas, supported by the implementation, for effective computations with ideals in the free algebra by utilizing the generalized letterplace correspondance. In particular, we provide a direct algorithm to compute Gröbner bases of non-graded ideals. Surprizingly we realize its behavior as "homogenizing without a homogenization variable". Moreover, we develop new shift-invariant data structures for this family of algorithms and discuss about them. Furthermore we generalize the famous criteria of Gebauer-Möller to the non-commutative setting and show the benefits for the computation by allowing to skip unnecessary critical pairs. The methods are implemented in the computer algebra system Singular. We present a comparison of performance of our implementation with the corresponding implementations in the systems Magma [BCP97] and GAP [GAP13] on the representative set of nontrivial examples.